Environmental Engineering Reference
In-Depth Information
to a decrease in surface energy of the solid. In principle, however, there is a related
term called adhesion tension,
A
sw
= σ
sa
− σ
sw
(s stands for solid), which can be used
to replace the interfacial tension term. The solid-air interfacial tension is assumed to
be a constant, and hence d
A
sw
=−
d
σ
sw
appears in the Gibbs adsorption equation.
3.5.2 E
QUILIBRIUM
A
DSORPTION
I
SOTHERMS AT
I
NTERFACES
The equilibrium between the surface or interface and the bulk phase is determined
by the same general principle of equality of fugacity (or chemical potential) as we
discussed earlier for bulk phases. Some common types of adsorption equilibria in
environmental engineering and their examples are shown in Figure 3.11.
At equilibrium, we can write
f
i
f
i
f
i
for interface/liquid equilibrium and
f
i
=
=
for interface-gas equilibrium. The superscript
represents the interface. In either
case this gives a relation between the surface concentration
Γ
Γ
i
and the bulk-phase
concentration
C
i
(or activity
a
i
)
. This relationship is called an
adsorption isotherm
,
andisaconvenientwaytodisplayexperimentaladsorptiondata.Toutilizetheequality
of chemical potentials, we need to obtain an expression for
f
i
Γ
i
. Let us
consider the surface as a two-dimensional liquid phase with both solvent molecules
and solute molecules in it. The solute
i
which has an excess concentration on the
surface is assumed to be dilute. The appropriate expression for fugacity is then
in terms of
f
i
= θ
i
γ
i
f
Γ
0
,
(3.74)
i
max
i
where
θ
i
is the fractional surface coverage of solute
i
, that is,
θ
i
= Γ
i
/
Γ
, with
max
Γ
i
is the maximum surface excess for solute
i
. Note that the standard fugacity is
defined as the chemical potential of the adsorbate when
1. It has to be so defined
since the liquid near the surface is different from that of the bulk liquid. Note that
if the surface phase was considered nonideal, then we need to replace the
θ
i
=
Γ
θ
sin
Gas
Gas
Liquid
Liquid
Solid
Solid
f
i
g
=
f
i
Γ
f
i
g
=
f
i
Γ
=
f
i
l
Air-sea
Air bubble in water
Fog in air
Rain drops in air
f
i
l
=
f
i
Γ
Aerosol in air
Soil-air
Activated Carbon
Soil-water
Sediment-water
Activated carbon-water
Metal oxid in water
Colloids in water
FIGURE 3.11
Examples of adsorption equilibria encountered in natural and engineered
environmental systems.
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